Suppose I am considering two hypotheses, *H*_{1} and *H*_{2}, and according to
*H*_{2} there are more
people. Does the fact that I exist give me reason to prefer *H*_{2}, all other things
being equal? If so, then my existence is apt to confirm the existence of
a multiverse over a single universe.

Here is one reason to think this works. The probability that I exist in a given world, all other things being equal, seems proportional to the number of people in that world. Each person in that world corresponds to another opportunity for me to exist.

While this is tempting, here is a toy model that should give us
pause. Suppose that I am defined by a real number parameter between
0 (inclusive) and 1 (not inclusive). According to hypothesis
*H*_{1}, a single real
number is picked uniformly at random in the range, and the person with
that parameter is created. According to hypothesis *H*_{2}, two real numbers are
picked uniformly and independently in the range, and persons
corresponding to these are created. Learning that a person with
*my* parameter is created seems to provide me with evidence for
*H*_{2}, since it’s
twice as likely on *H*_{2} as on *H*_{1}.

But this is tricky. In classical probability theory, it is correct to
say that my parameter is twice as likely to be generated on *H*_{2} as on *H*_{1}, but that’s only
because both probabilities are zero, and zero is twice zero, so while
*H*_{2} is twice as
likely as *H*_{1}, it is
also true that *H*_{1}
is twice as likely as *H*_{2}!

Perhaps, though, we want to depart from classical probability theory
in some way, say by allowing non-zero infinitesimal probabilities or by
an intuitive handwavy “this is twice as likely as that”. However, it is
then no longer clear that on *H*_{2} there is twice as big
a chance of hitting my parameter. For there are (infinitely) many ways
of picking a number between 0 and 1 uniformly randomly.

Here’s one way:

- You write down “0.”, then roll a fair ten-sided die infinitely many times, writing down the results as the digits after the decimal point, thereby generating a decimal representation of a number. If the number ends with infinitely many nines, try again.

(The final proviso is to ensure that intuitively each number is equally likely. Without that proviso, 1/10 would be more likely than 1/3, as there would be two ways of getting 1/10, namely 0.1000... and 0.0999..., but only one way to get 1/3, namely 0.3333.....)

Here is another way:

- You write down “0.”, then roll a fair ten-sided die infinitely many times, omitting the results of the first die throw, but writing down the results as the digits after the decimal point, thereby generating a decimal representation of a number. If the number ends with infinitely many nines, try again.

Intuitively, method B has ten times the probability of generating any
given number than method A has, as long as literally the numerically
same die throws occur in the two cases. For consider the number 1/3 = 0.3333.... By method A to generate it
you need every die to show a three. By method B, to generate 1/3, all you need is for all the die throws
*other than* the first one to be threes, and so there are ten
times as many ways to generate the number.

Now, if the single selection of a parameter on *H*_{1} uses method B while
the double selection of a parameter on *H*_{2} uses method A, then
intuitively we are five times as likely to generate my parameter on
*H*_{1} than on *H*_{2}. Thus merely saying
that on both hypotheses the parameters are generated uniformly is
insufficient to determine how the comparison between the probabilities
of generating my parameter goes.

We might insist that in both hypotheses *the same* method for
generating parameters is used. But notice that in cosmological
applications, this is implausible. If *H*_{2} is some multiverse
hypothesis and *H*_{1}
is a single universe hypothesis, we are unlikely to be able to count on
the two hypotheses involving even the same laws of nature, much less the
same selection process for the parameters of the persons. (Besides all
this, it is really unclear what it even counts to say that there are two
different runs of method A.)

So, here’s what I am thinking. On classical probability theory, there
is no difference in the probability of my parameter getting generated on
*H*_{2} than on *H*_{1}, because both
probabilities are zero. On non-classical probability theory, we can
perhaps make sense of a difference between the probabilities, but cannot
count on the hypothesis with more people being more likely to generate
my parameter.

Given all this, there does not seem to be a way of making sense of comparing the evidential impact of my existence on the two hypotheses using probabilistic methods. Maybe all we have is intuition.

## 2 comments:

Lots of worries about this…

The main worry is that ‘my’ parameter is being selected after the event. If someone had specified my parameter before the event, and then found that someone with my parameter had been created, that could (arguably, as in the post) favour H2. But if I simply note my parameter, nothing useful follows, because anyone who was created would have

someparameter. The only thing ‘unspecified-my’ existence does is (trivially) rule out hypotheses in whichno onecould have been created.I don't worry about my parameter being selected after the event. The parameter is what it is. We can conditionalize on it, just as we can conditionalize on the outcome of a spinner even if we didn't think about the outcome before we spun the spinner.

My more serious worry is this. In the toy case of a numerical parameter, it seems that the right way to handle the situation is to say: We can't conditionalize on the exact value of the parameter, so we conditionalize on some range of parameter values with non-zero measure. And then we get clear and unambiguous support for H2. More in today's post.

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